Journal of Glaciology

Article

Cite this article: Meehan TG, Marshall HP,Bradford JH, Hawley RL, Overly TB, Lewis G,Graeter K, Osterberg E, McCarthy F (2021).Reconstruction of historical surface massbalance, 1984–2017 from GreenTrACSmulti-offset ground-penetrating radar. Journalof Glaciology 67(262), 219–228. https://doi.org/10.1017/jog.2020.91

Received: 10 June 2020Revised: 1 October 2020Accepted: 2 October 2020First published online: 14 December 2020

Keywords:Glacier geophysics; ground-penetrating radar;polar firn; surface mass budget

Author for correspondence:Tate Meehan,E-mail: tatemeehan@u.boisestate.edu

© The Author(s), 2020. Published byCambridge University Press. This is an OpenAccess article, distributed under the terms ofthe Creative Commons Attribution licence(http://creativecommons.org/licenses/by/4.0/),which permits unrestricted re-use,distribution, and reproduction in any medium,provided the original work is properly cited.

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Reconstruction of historical surface massbalance, 1984–2017 from GreenTrACSmulti-offset ground-penetrating radar

Tate G. Meehan1,2 , H. P. Marshall1,2 , John H. Bradford3, Robert L. Hawley4 ,

Thomas B. Overly5,6 , Gabriel Lewis4 , Karina Graeter4 , Erich Osterberg4

and Forrest McCarthy4

1Department of Geoscience, Boise State University, Boise, ID, USA; 2U.S. Army Cold Regions Research andEngineering and Laboratory, Hanover, NH, USA; 3Department of Geophysics, Colorado School of Mines, Golden,CO, USA; 4Department of Earth Sciences, Dartmouth College, Hanover, NH, USA; 5Cryospheric Sciences Lab, NASAGoddard Space Flight Center, Greenbelt, MD, USA and 6Earth System Science Interdisciplinary Center, University ofMaryland, College Park, MD, USA

Abstract

We present continuous estimates of snow and firn density, layer depth and accumulation from amulti-channel, multi-offset, ground-penetrating radar traverse. Our method uses the electromag-netic velocity, estimated from waveform travel-times measured at common-midpoints betweensources and receivers. Previously, common-midpoint radar experiments on ice sheets havebeen limited to point observations. We completed radar velocity analysis in the upper ∼2 m toestimate the surface and average snow density of the Greenland Ice Sheet. We parameterizedthe Herron and Langway (1980) firn density and age model using the radar-derived snow density,radar-derived surface mass balance (2015–2017) and reanalysis-derived temperature data. Weapplied structure-oriented filtering to the radar image along constant age horizons and increasedthe depth at which horizons could be reliably interpreted. We reconstructed the historical instant-aneous surface mass balance, which we averaged into annual and multidecadal products along a78 km traverse for the period 1984–2017. We found good agreement between our physicallyconstrained parameterization and a firn core collected from the dry snow accumulation zone,and gained insights into the spatial correlation of surface snow density.

1. Introduction

The Greenland Ice Sheet (GrIS) expresses high variability in ice loss, and hence sea level rise,due to the regional scale variability in the processes governing mass balance (Lenaerts andothers, 2019). Surface mass balance (SMB) contributes just over half (�52%) of GrIS massloss, but ice-sheet wide SMB simulated from regional climate models maintains �25% uncer-tainty (Shepherd and others, 2020). Efforts to improve SMB simulation (e.g. Fettweis andothers, 2017) are limited by the scarcity of observations, which are required to evaluate themodel performance (e.g Noël and others, 2016). Traditionally, SMB measurements are madeat the point scale during infrequent field efforts, through the laborious process of excavatingsnow pits or drilling firn cores. The sparseness of snow pit observations on the GrIS limits thetestable correlation lengths and tends to debilitate spatial correlation analysis. Consequentially,surface density measurements have shown no spatial correlation over length scales of tens tohundreds of kilometers (Fausto and others, 2018). Due to the unknown variability of densityand SMB, point measurements used to parameterize a firn model (e.g. Zwally and Li, 2002)must be extrapolated to regional scales cautiously. In space-borne altimetry retrievals ofGrIS mass balance, the uncertainty in modeled corrections for snow densification requiredto convert a measured change in ice-sheet volume to a change in mass causes �16% uncer-tainty (Shepherd and others, 2020).

Ground-penetrating radar (GPR) surveys are capable of imaging layers of accumulatedsnow (e.g. Vaughan and others, 1999). However, conventional, single-offset GPR analysisrequires an independent measurement of firn density to estimate the accumulation(Navarro and Eisen, 2009). Point SMB measurements often provide the required density infor-mation to extrapolate the density profile along the track of the radar sounding (e.g. Hawleyand others, 2014; Overly and others, 2016). Yet, relying on sparse firn cores to extrapolatedensity over tens to hundreds of kilometers may bias the derived accumulation estimates.For example, ice lenses sampled in a firn core increase the average density and can beincorrectly extrapolated over tens of kilometers, as these features are uncorrelated over tensof meters (Brown and others, 2011). For the period 1971–2016, greater than 10% bias tothe SMB is possible, when firn cores are not available for extrapolation (Lewis and others,2019). Inaccuracies are greater in southern Greenland, which is experiencing increasingnear-surface firn densification as a result of atmospheric warming (Graeter and others,2018), than in central Greenland. Parameterization of snow and firn densification continuesto improve (e.g. Meyer and others, 2020); yet, evolving the firn using full energy balance

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modeling remains operationally challenging and is limited spa-tially by the unknown heterogeneities of surface snow density,accumulation and melt (Vandecrux and others, 2018). Surfacesnow density parameterizations formulated around temperatureand wind speed (e.g van Kampenhout and others, 2017) arearguably less preferable than density measurements because ofuncertainties in estimating wind speed and modeling theunknown length scale variability that exists in the GrIS snow(Fausto and others, 2018).

Radar retrievals of snow density are an appealing alternativeto in situ observations of snow and firn because the methods arenon-destructive and rapidly acquire vast amounts of data.However, few methods for continuously mapping snow andfirn density exist (e.g. Grima and others, 2014) due to the com-plexities of data inversion. In this work, we present the analysisof multi-channel, multi-offset, radar (MxRadar) imagery along a78 km traverse in the GrIS dry snow accumulation zone to dem-onstrate the capability of this method, which has the advantageof ascertaining snow and firn density, and depth, and therebySMB, independently. Of the previous studies applying GPR vel-ocity analysis, none have performed continuous estimatesthroughout tens of kilometers distance (e.g. Bradford and others,2009). We based our MxRadar workflow on the analysis of theradar surface wave, which exhibits linear moveout (LMO), andthe fall 2014 isochronous reflection horizon (IRH) to estimatethe surface snow density, column average density, horizondepth and 2015–2017 SMB. We then input our data into theHerron and Langway (1980) firn density and age model. Weuse the firn model to further enhance the MxRadar imageryand extend the historical period of the SMB reconstruction to1984–2017 with instantaneous (∼14 days) temporal intervals.We compare the resulting SMB against a firn core and quantifythe length of spatial correlation that exists in surface snowdensity. We quantify the bias reduction in SMB derived usingthe measured-modeled, MxRadar–Herron and Langway (1980)method. Then we provide a discussion of the results, limitationsand advantages of the method, and future directions. We devel-oped our analysis within the interior region of Greenland wherethere was significant spatial variation in accumulation, but littlemelt, to develop confidence in this type of radar retrieval fordensity and SMB.

2. Greenland Traverse for Accumulation and Climate Studies

The Greenland Traverse for Accumulation and Climate Studies(GreenTrACS) is a multi-disciplinary study of recent SMBchanges in the West Central percolation and dry snow accumula-tion zones of the GrIS. During the Spring of 2016 and 2017, wetraveled a total of 4436 km by snowmobile from Raven/DYE-2to Summit Station along the elevation contour straddling thepercolation zone, and along West-East ‘spurs’ perpendicular tothe elevation contours. Throughout the expedition, we collected16 shallow (22–32 m) firn cores and dug 42 snow pits; 16 pitswere coincident with the cores and the 26 others were dug atthe ends of the spurs (Figs 1 and 2). Our GreenTrACS field sea-sons occurred prior to the on-set of melt to reduce the complexityof radar data inversion. The cores and the coincident snow pitswere sampled for density, isotopic chemistry, dust and traceelements to define annual layer depths for measuring SMB (e.g.Graeter and others, 2018; Lewis and others, 2019). As firn coresare strategically located point measurements, GPR imagery isoften leveraged to spatially extend the record of firn stratigraphybetween core sites for accumulation studies (e.g. Spikes andothers, 2004; Miège and others, 2013). We operated a suite ofradar instruments spanning the frequency range 0.4–18 GHz;the focus of this study is the MxRadar.

2.1. Study area

GreenTrACS Core 15 (GTC15) is the second most northern coresite of the GreenTrACS campaign (47.197°W, 73.593°N) and is∼2600 m above sea level. GTC15 had an average annual tempera-ture of −25.7 ± 1.0°C (Modern-Era Retrospective analysis forResearch and Applications (MERRA), 1979–2012), and an averageannual SMB of 0.306 ± 0.021 mw.e. a−1 (1969–2016). The siteexperiences little to no melt, measured as the average melt featurepercentage determined by normalizing each year’s ice layer waterequivalent by the annual water equivalent and then averaging(0.47%, 1969–2016).

GTC15 Spur West is a triangular, clockwise circuit that departsfrom and returns to GTC15 (Fig. 1 inset). The first of three trans-ects is 15 km in length, bearing 157°, and begins at GTC15. Thesecond transect is 30 km in length at 246.5° which ends at Pit15W. The final transect is 33 km in length from Pit 15W toGTC15 and bearing 40.5°. The GrIS surface of GTC15 SpurWest was wind-affected snow with sastrugi 25 cm in height. Weestimated the average meteorological wind direction of 152°using monthly 10 m zonal and meridional wind speeds fromthe ensemble of the third generation reanalysis models(GEN3ENS) for the period 1979–2012 (Birkel, 2018). The averagewind direction is approximately parallel to the first transect ofGTC15 Spur West, approximately orthogonal to the second tran-sect, and 21.5° oblique to the third transect. The cyclicity in thetopographic profile (Fig. 2) results from our return to GTC15along a path oblique to the path approaching Pit 15W. TheSMB changes significantly across the 5 km wide trough betweendistances 40–50 km. But, we do not observe preferential windwardand leeward affects to the accumulation pattern here, because theorientation of the transects crossing this topographic trough areapproximately orthogonal to the average wind direction. Weselected this particular spur to develop our processing and ana-lyses because of the apparent interplay between the surface eleva-tion, SMB and heterogeneous layering observed in the radarimagery. Yet, we have foregone any topographic corrections inthe radar processing.

2.2. Field methods

The MxRadar is a Sensors & Software 500 MHz GPR deployedwith a multi-channel adapter in a multi-offset configurationusing three transmitting and three receiving antennas (Fig. 3).During data acquisition, the transmitting and receiving channelswere multiplexed to form nine radargrams which have independ-ent antenna separations (offsets). The antennas were co-polarized,perpendicular to the direction of travel, and all are specified at500 MHz with greater than two octave bandwidth. However,dependent on the antenna pairing, the actual central frequencyand bandwidth varied on the order of tens of MHz. Our methodsand analysis are tailored to produce meaningful data for theevaluation and improvement of snow cover and firn models andregional climate and reanalysis modeling of SMB.

3. Analysis methods

We review multi-offset GPR methods for SMB calculations inSection 3.1 to clarify the advantages of the multi-offset techniquethat are also important for interpreting the results. We providemuch of the methodological detail in the SupplementaryMaterial S.1. Here, we touch on the methodology to simplifyour strategy for reconstructing the historical SMB for the period1984–2017 along GTC15 Spur West. We considered SMB ratherthan the accumulation rate because of unaccounted mass lost tosublimation and ablation. SMB is conventionally measured

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using GPR by interpreting a select few IRHs using a constant ageinterval and applying the average normalized firn density overthis interval (e.g. Lewis and others, 2019). Instead, we relied onthe models of density and age, which were discretized in depthat a comparable resolution to the GPR data, and generated aSMB model with instantaneous (∼14 day) temporal intervals(Section S.1.3). We averaged annual SMB from many realizationsof the instantaneous SMB model in a Monte Carlo simulation toassess uncertainty (Section S.1.4). We estimated the multidecadalaverage SMB, invoking the central limit theorem, by repeatedly

drawing from 10 of the 33 annual SMB distributions at randomand averaging.

To parameterize the firn model, we first completed conven-tional signal processing on the nine radargrams, which consistedof a two octave bandpass filter around 500 MHz, amplitude gaincorrections for wavefront spreading, coherent noise removal(background subtraction) and random noise removal (smooth-ing). Then we interpreted the air wave, surface wave and a shallowreflection (Fig. 4) on each of the nine images using a semi-automatic picking algorithm (Section S.1.1). We inverted the

Fig. 1. GreenTrACS firn cores (GTCs) are numbered 1–16. Ground-penetrating radar surveys were conducted along spur traverses and the main route that linksthe GTCs. We developed our radar processing and analyses at GTC15 Spur West (lower left inset). The 2000 m asl contour envelopes the western spurs. Surfaceelevation was acquired from Morlighem (2017) and Porter and others (2018).

Fig. 2. Topographic profile of GreenTrACS Core 15 Spur West. The topographic undulation near Pit 15 W is responsible for increases and decreases in accumulation.The initial 15 km, up to the point of maximum elevation of the profile, are directed into the predominant wind, making this a leeward slope. The predominant windblows approximately orthogonal across the next 30 km of the GTC15 Spur west traverse and is 21.5° oblique to the final 33 km of GTC15 Spur West.

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travel-times of the surface wave and the shallow reflection (seeSection 3.1.1) to estimate the average electromagnetic (EM)propagation velocity and depth of the dry snow and firn in aleast-squares approach (Section S.1.2), which used random resam-pling of the data to estimate uncertainties (Section S.1.4). Wethen applied a petrophysical model (Wharton and others, 1980)which relates the EM velocity of dry snow and firn to its density(Section S.1.3).

Our measured-model approach relied on the Herron andLangway (1980) empirical firn density and age model, hereafterHL, which requires three input parameters: average snow density,average annual accumulation and 10 m firn temperature. Weparameterized the HL model with the MxRadar snow density,MxRadar SMB (2015–2017) and MERRA 2 m air temperatureas a proxy for firn temperature (Loewe, 1970), to model the strati-graphic age and density of the firn. We assessed the firn modelaccuracy and sensitivity to parameterization to illustrate theaccuracy of the MxRadar-HL (MxHL) firn density (SectionS.1.5). We justified tuning the age model to improve our estimatesof SMB in a process that jointly updated the age-depth and SMBmodels according to the radiostratigraphy. The age model allowedus to convert the time domain radar image into the stratigraphicage domain, known as the Wheeler (1958) domain. In principle,the firn structure can be estimated by the age model because thestratigraphy was deposited in isochronous layers. The imaged firnstructure can be flattened by converting the time domain GPRimage into the Wheeler domain because the rows of theWheeler image maintain a constant age. We ensured the relativestructure of the age model by picking five horizons of the Wheelertransformed radiostratigraphy with an average epoch of 5.3 ± 2.7years (the latest being the 1991 horizon) and perturbing the agemodel with the interpolated residuals to re-flatten the Wheelerimage. We developed a structure-oriented noise-suppression filterwhich operates along the radar reflection horizons in the Wheelerdomain to eliminate remnant noise after conventional GPR signalprocessing (Section S.1.6). This innovative signal processing tech-nique allowed SMB estimates to depths at which previously thestratigraphy was uninterpretable due to the low signal-to-noiseratio. We then converted the filtered radargram from theWheeler domain into the depth domain and interpreted 16IRHs with an average epoch of 2.1 ± 1.7 years dating back to1984. We calculated the error between the GTC15 geochemicallydetermined age-depth scale and the 16 picked IRHs and interpo-lated a second grid of perturbations which we applied as a finalupdate to the age model. We calculated the instantaneous SMBby taking a numerical derivative of the age-depth model (dz/da)and multiplying it by the MxHL density model (Eqn (S.20)).

3.1. Review of multi-offset radar

Common-midpoint (CMP) radar surveys are practiced in glaci-ology to estimate the EM wave speed of the ice, air and/or

water mixture (e.g. Eisen and others, 2002). The wave speed isrelated to firn density and liquid water content using a dielectricmixture formula for a two or three phase relationship (e.g.Looyenga, 1965; Wharton and others, 1980). In most studies,the CMP survey is treated as a point measurement of the firn ver-tical density profile, which is less laborious than extracting a core,but offers less vertical resolution and accuracy. Prior toGreenTrACS, CMP experiments on ice sheets were limited topoint observations. We synthesized continuous CMP data by tow-ing a streamer of nine antenna pairs that were linearly spacedfrom 1.33 to 12 m apart (Fig. 3). While the antenna pairs inthis deployment did not have a common midpoint, we rebinnedthe constant offset radargrams for each pair independently,such that the analysis was performed on offset gathers with com-mon midpoints.

3.1.1. Interpreting the near-surface wavesNumerous geophysical methods exist for velocity analyses ofCMP data gathers. Analyses of reflection data can be dividedinto two fundamental categories by the question, ‘Does the ana-lysis assume normal moveout?’ Normal moveout (NMO) is thereflection travel-time dependence on offset that arises from ahomogeneously-layered and planar subsurface structure (withinthe distance of the maximum antenna offset) that exhibits smallvertical velocity heterogeneity (Al-Chalabi, 1974). Previous stud-ies avoided classical NMO analysis, instead using less automated,more computationally expensive methods that favored accuracy(Bradford and others, 2009; Brown and others, 2012, 2017).Many caveats of NMO velocity analysis and sources of error inthe radar common-midpoint analysis are discussed in Barrettand others (2007). We demonstrate that NMO analysis of thesnow and shallow firn yields a satisfactory result for data withlow noise (see Supplement S.1.5), as ice-sheet stratigraphy inthe high elevation accumulation zone is close to homogeneousand planar at the length scale of the radar streamer array.

Linear moveout (LMO) is the one-way travel-time dependenceon offset of radar waves traveling directly from the transmitterthrough the air over the ice sheet and through the snow underthe ice-sheet surface to the receiver antenna. We assumed that theair wave expresses the linear moveout velocity c≈ 0.2998 m/ns tocalibrate the timing of the multi-channel system (Section S.1.2).To analyze the surface wave, we assumed that the shallow, surficialsnow is also planar and homogeneous at the scale of the maximumoffset. We identified the air wave, surface wave and a near surfacereflection and their respective moveout behavior in Figure 4. Thetravel-times of these waves were interpreted using a horizon trackingalgorithm (see Supplement S.1.1). The linear methods for LMO andNMO velocity analysis are described in Section S.1.2 and the meth-ods for estimating the surficial and average snow density and depthof the fall 2014 IRH are discussed in Section S.1.3. We quantifiedthe uncertainty of the density, depth, age and SMB used to param-eterize the HL model in Section S.1.4.

Fig. 3. The MxRadar streamer array has three transmitting (Tx) and three receiving (Rx) antennas, which form nine independent offsets that were linearly spacedfrom 1.33 to 12 m apart. We simultaneously acquired nine continuous radargrams (one for each constant offset) and then binned the source-receiver pairs intocommon-midpoint (CMP) gathers.

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3.1.2. Critically refracted wavesLateral energy travels on direct raypaths from transmitter toreceiver, but also on raypaths that are critically refracted at thefree-surface. An upgoing reflected wave can become criticallyrefracted along the air/snow interface upon exiting the snow sur-face. These refracted waves appear in Figure 4 as multiple air wavearrivals succeeding the initial air wave. In Section S.1.2.1, we pro-vide a discussion of the critically refracted wave phenomena withan accompanying snowpack model and exercise to support anddemonstrate the critically refracted raypath.

3.2. Spatial correlation of surface snow density

The LMO and NMO estimated snow densities are independentmeasurements of the snow density above the interpreted radarhorizon. The GPR surface wave maintains a fairly consistentdepth level (∼0.5 m, Eqn (S.17)), but the NMO reflection horizondoes not. To mitigate the effects of depth on the correlation, weextracted the rows of the MxHL density model correspondingto the average depth of the LMO (0.5 m) and NMO (1.92 m) hor-izons interpreted for velocity analysis (Fig. 4). We used Pearson(1907) correlation to determine the relationship between thedensity at 0.5 m depth and the density at 1.92 m depth.Additionally, we conducted variogram analysis (Matheron,1963) on the LMO estimated snow density for each of the threetransects of GTC15 Spur West. We determined the length scaleover which there is consistent spatial correlation of the surfacesnow density across all three transects as the distance where thethree experimental variograms diverge. We understand this diver-gence point as the experimental range of the variogram with theshortest length scale of correlation. We determined the experi-mental range of the remaining two variograms at the seconddivergence point and as a significant slope break or change inconcavity/convexity.

4. Results

The multi-offset radar travel-time inversion determined the GrISsurface snow density and average snow density without manualobservations (Fig. 5). We estimated the 2015–2017 SMB fromthe MxRadar-derived snow depth and density using the GTC15age of the near-surface IRH (Fig. 5). The LMO and NMOdensities were independently estimated and strongly correlate(R2 = 0.67, p = 0). Spatial patterns in the LMO derived snow dens-ity are consistent for three azimuths up to 2 km lag distance(Fig. 6). The multidecadal average 10 m wind direction fromGEN3ENS (1979–2012) along GTC15 Spur West is approxi-mately 152°. With information on the predominant wind direc-tion, a closer look at Figure 6 reveals directionality in the spatialpattern of surface snow density. The range of the variogram forthe 157° transect (in the direction of the predominant wind) is6 km, the range of the 246.5° transect (orthogonal to the predom-inant wind) is ∼2 km, and the range of the 40.5° transect (obliqueto the predominant wind) is ∼3 km. By combining the radar-derived density and SMB with MERRA 2 m temperature, weaccurately parameterized the HL firn density and age model.For depths up to ∼22.5 m, the mean absolute error betweenGTC15 densities and MxHL densities is 9.6 kg/m3, with a biasof 1 kg/m3, and rms error of 12.2 kg/m3. We find that extrapolat-ing the GTC15 densities along GTC15 Spur West introducesan insignificant (on the order of 1%) bias to the SMB of−0.004 m w.e. a−1 and rms error of 0.005 m w.e. a−1. The MxHLfirn model permitted radar imaging in the depth and stratigraphicage domains. In Figures 7 and 8, we illustrate our structure-oriented filter along GTC15 Spur West between 35 and 55 km dis-tance, where the largest heterogeneity in firn stratigraphy occurs.After applying structure-oriented filtering, we were able to inter-pret significantly more IRHs and refine the age-depth model to anaccuracy of ±31 days (see Supplement S.1.4). We reconstructedthe temporal SMB history from January 1984 to January 2017

Fig. 4. This offset gather is represented by radargrams recorded at offsets 4, 8 and 12 m along the initial 45 km of GTC15 Spur West, and is annotated to convey thewaveforms used in our analysis and the concepts of normal moveout (NMO) and linear moveout (LMO). Consider the traces at zero distance for each offset as aCMP gather. The air wave and surface wave arrivals are modeled by a linear expression of travel-time as a function of offset (Eqn (S.1)). The air wave is the first toarrive and expresses a more shallow slope (faster velocity) than the surface wave which is impeded while traveling through the snow. The annotated reflectionexpresses non-linear moveout which is approximated by NMO (Eqn (S.2)). The surface-wave (LMO) and reflection (NMO) annotated in this diagram are used toestimate the surface snow density, average snow density and depth of the fall 2014 isochronous reflection horizon (IRH). The age of the horizon was determinedat GTC15 and allowed us to estimate the 2015–2017 SMB (see Supplement S.1.3), and in turn, is used to parameterize the HL model (see Supplement S.1.5).

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and compare our result to the GTC15 firn core-derived SMB inFigure 9. The MxHL SMB history has a mean absolute error of0.038 m w.e. a−1, a bias of 0.004 m w.e. a−1 and an rms error of0.047 m w.e. a−1. Uncertainty in the SMB measured fromGTC15 was calculated following Graeter and others (2018).Average uncertainty in annual SMB is 0.036 and 0.044 mw.e. a−1

for MxHL and GTC15, respectively. The mean thickness of anannual layer for the period 1984–2017 is 57.9 cm as measured atGTC15. The mean absolute error in the thickness of anannual layer estimated by MxHL is 7.8 cm, which contributes0.039 mw.e. a−1 (13%) error in the SMB reconstruction on average.Density inaccuracies in the SMB reconstruction result in a0.004 mw.e. a−1 (1.3%) error on average. The MxHL 1984–2017

multidecadal average SMB is 0.297 ± 0.016 mw.e. a−1 and is agood estimator of the GTC15 1984–2017 multidecadal averageSMB (0.301 ± 0.025 mw.e. a−1). At GTC15 the 2015–2017 averageSMB is within the uncertainty bounds of the multidecadal averagesspanning 1969–2017, the oldest period spanned by the core, and1984–2017 the period spanned by the MxRadar imagery.

5. Discussion

We independently assessed the four sources of uncertainty in theMxHL SMB (depth, density, temperature and age) and then pro-pagated these uncertainties through the MxHL model by MonteCarlo simulation to estimate the SMB mean and standard devi-ation for each year of 1984–2017. On average, the differencebetween GTC15 and MxHL SMB is small enough to acceptthe MxHL measured-modeled densities in place of extrapolatingthe measured firn core density along GTC15 Spur West.Extrapolated densities are likely to be much less accurate fartherfrom core sites and in the percolation zone, due to increased near-surface pore space reduction caused by melt water infiltration(Harper and others, 2012). We also expect the accuracy of theHL density model to break down at elevations within the perco-lation zone (Brown and others, 2012). Annual fluctuations indensity, and density excursions due to warming events, are notcaptured in the HL model. Using the MxRadar, we have the abil-ity to measure the density profile in the percolation zone withadditional layer picking for near-surface velocity analysis, butthe NMO approach is sensitive only to the average density ofintervals in between the layer picks (Dix, 1955) and is susceptibleto errors due to subsurface velocity heterogeneities and data noise(Al-Chalabi, 1974).

In the upper ∼2 m of the firn column, we replaced modeleddensities with a linear fit between the two radar measurementsof snow and firn density using the surface wave and the reflectionfrom the fall 2014 IRH. This reduced the near-surface bias pre-sent in the HL density profile and we found strong correlationbetween the densities of these independent radar measurements.The richness of the MxRadar data stream permits geostatisticalanalysis at the sub-kilometer scale. We found that local (on theorder of 1 km neighborhood) processes control the GrIS drysnow density. The similarity in spatial patterns of radar estimatedsurface snow density, up to ∼2 km lag distance, contrasts withthe findings that no correlation exists between surface snow dens-ity, latitude, longitude or elevation (Fausto and others, 2018),which is likely due to the limited observations of snow densityat the <1 and <10 km scales within the Surface Mass Balanceand Snow Depth on Sea Ice Working Group dataset(Montgomery and others, 2018). Our variogram analysis wastested to 15 km lag separations along three azimuths: 157°,246.5° and 40.5°. In the direction of the prevailing wind, wefound 6 km correlation distance with diminishing correlationlength for transects increasingly orthogonal to the prevailingwind. We found that the SMB decreased with increasing slopeon the leeward, 157°, transect, which corroborated the findingsof Arcone and others (2005). We did not find trends in SMBwith slopes of the 246.5°, and 40.5° transects, as these transectswere approximately orthogonal and 21.5° oblique to the predom-inant wind direction, respectively. Future application of thismethod to the 4000 + km traverse will allow the exploration ofsurface density variations at much larger scales and at additionalorientations relative to the prevailing winds.

The 2014–2017 SMB appears to be overestimated by MxHL,though the near-surface radar velocity analysis was focused onthis range. We support the radar findings here with the under-standing that firn samples recovered from these depths are sus-ceptible to in situ losses due to their unconsolidated nature.

Fig. 5. The MxRadar inversion parameter distributions along GTC15 Spur West. TheLMO and NMO densities were independently estimated and strongly correlate (R2 =0.67, p = 0). The MxHL model is parameterized by the average of the LMO andNMO densities, the 2015–2017 average SMB and MERRA (1979–2012) average 2 mtemperature.

Fig. 6. We calculated experimental variograms of the LMO estimated snow densityalong the three azimuths of GTC 15 Spur West using lag separations up to 15 km.Plotted in log-log space, the linearity of each variogram slope indicates that spatialcorrelation among the three azimuths exists up to ∼2 km distance. Correlationbeyond this distance is difficult to assess given the limited azimuths and lag separa-tions possible for GTC 15 Spur West. However, predominant wind direction appearsto have a control on the correlation length, as evidenced by the 6 km range of the157° transect variogram (in the direction of the predominant wind) and shorter,∼2 and ∼3 km ranges of the 246.5° transect (orthogonal to the predominant wind)and 40.5° transect (oblique to the predominant wind), respectively.

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The radar retrieval has a sample footprint of approximately∼25 m (twice the length of the antenna array) and is non-destructive, while the borehole diameter is ∼8 cm and samplesonly one point in space. It is also likely that the age model isless accurate nearest the ice-sheet surface due to core sampleloss; however, we sacrifice greater accuracy in the radar domainbecause of the limitations in our ability to interpret depthimage. The fall 2014 horizon was the latest IRH measured inour analysis. Picking annual reflection horizons later than 2014,near the model boundary, created steep gradients in the numericalderivative required to estimate the SMB which yielded erroneousvalues.

We see evidence of the 2012 melt event (Nghiem and others,2012) in the filtered depth image (Fig. 8). At 3 m depth, the top ofthe reflection sequence represents January 2013, and at 4 m depth,the bottom of the sequence is January 2011. This IRH sequenceexpresses fading and discontinuity that, we hypothesize, is theresult of 2012 melt water infiltration. Measured at GTC15, the

2011 annual layer has a melt feature percentage of 7.9%.However, melt water-induced firn densification does not explainthe inaccuracy in 2010 MxHL SMB, as 2010 recorded 0% meltfeature percentage at GTC15. The MxHL density model is accur-ate within the 2010 annual layer, rather our estimate of the 2010annual layer thickness is 22 cm thinner than measured at GTC15.This is the second largest error in annual layer thickness, onlybehind the 2015 layer which was estimated to be 24 cm thickerthan measured at GTC15 because of the aforementioned issuesin estimating SMB near the model boundary. The degradedimage quality of the 2011–2013 IRH sequence inhibited our abil-ity to interpret the age sequence accurately enough to define theannual layer thicknesses for 2011 and 2012. Instead, we reliedon interpolation to approximate the thickness of these horizons.The leading source of error in the historical SMB reconstructionare inaccuracies in the age model that result from limitations inour ability to interpret the radar image, even after applying thestructure-oriented filter.

Fig. 7. Conventional GPR processing was applied to each of the nine constant offset radargrams. We then performed NMO correction to project each constantoffset image to zero offset. We stacked the NMO corrected radargrams together to synthesize one conventional GPR travel-time image. The travel-time imageremains quite noisy, and it is difficult to interpret due to the discontinuities along the reflection horizons.

Fig. 8. The travel-time image (Fig. 7) is first transformed into the stratigraphic age domain, known as the Wheeler (1958) domain. Then we applied structure-oriented filtering to the Wheeler domain image and converted into the depth domain. The depth section, taken from GTC15 Spur West, has remarkable continuityalong the reflection horizons, which allows us to interpret IRHs to ∼22.5 m depth. The undulation in the firn stratigraphy is caused by spatial variability in snowaccumulation. It is necessary to interpret along steeply varying undulations like these to evaluate high resolution (< 5 km) regional climate model simulations ofSMB. However, without the structure-oriented filter we would be unable to track the reflection horizons along the undulations.

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The multidecadal average SMB for the period 1984–2017 atGTC15 has remained nearly constant. Yet, sinusoidal variabilityin SMB on the decadal time scale is apparent in the MxHL histor-ical SMB reconstruction and is confirmed by GTC15 SMB. Decadalvariability in the MxHL reconstruction would not be observablewithout the application of structure-oriented filtering and inter-pretation that permitted an accurate instantaneous SMB model.For GPR imagery expressing small or gradual SMB variability, itmay be sufficient to apply the structure-oriented filter in theWheeler domain without the steps of interpretation, age model cor-rections and image re-flattening (Section S.1.6). The snow densityestimation component is unique to the multi-offset radar and inte-gral in our ability to parameterize the HL model. However, thestructure-oriented filtering can be applied to any GPR imagery ofisochronous firn, provided a stratigraphic age model in the radartravel-time domain is used as a proxy for the firn structure.

Along GTC15 Spur West, we expect the largest errors due tofirn advection to occur across the studied undulations (Figs 7and 8), where the SMB gradient is largest and oscillating. Thetwo undulations here represent the same feature observed on out-bound and inbound traverses, and serve as a demonstration of therepeatability of the methods. In regions where the spatial gradientin SMB is dynamic or ice-sheet surface velocities are large, theadvection of firn mass decreases the accuracy of radar estimatedSMB. On Pine Island Glacier, with ice surface velocities on theorder of 10–103m a−1, strain corrections applied to the accumula-tion model amounted to a 1% correction to the 1986–2014 aver-age SMB (Konrad and others, 2019). Ice surface velocities alongGTC15 Spur West are on the order of 10 m a−1 (Joughin andothers, 2018), and therefore we accept a contribution of errorthat is an order of magnitude less than the uncertainty, by notapplying corrections for the SMB due to advection.

6. Conclusions

GreenTrACS conducted the first multi-offset GPR traverse onthe Greenland Ice Sheet, covering a total distance of 4436 km.We examined a 78 km section of the GreenTrACS 2017 traverse(GTC15 Spur West) to develop the methodology for multi-offsetGPR wave velocity, imaging and uncertainty analyses to accuratelyquantify the surface snow density, average snow density, firndensity, instantaneous SMB, annual SMB and multidecadal aver-age SMB for the period 1984–2017. Using travel-time inversion ofthe radar waveforms, we continuously mapped Greenland snowdensity without manual observations of the snow. We found con-sistent spatial correlation of near-surface density for separationsup to 2 km distance and evidence to support the prevailingwind direction as a source of correlation up to 6 km distance.We found significant correlation (R2 = 0.67, p = 0) between near-surface snow density and average snow density of the upper 2 m.We demonstrated the use of the Herron and Langway (1980)model that was parameterized by the radar-derived snow density,radar-derived SMB (2015–2017), and MERRA 2 m air tempera-ture, to estimate firn density and age. Our measured-modeledfirn density in the dry snow accumulation zone accurately repre-sents the firn core but can be performed continuously along a tra-verse in the field without destructive measurements. GreenTrACSCore 15 Spur West presented an interesting challenge because ofspatial SMB variability that is enhanced by the surface topog-raphy. In the dry snow zone, the topographic effect induces undu-lations in the firn stratigraphy which steepen with depth, due tothe persistence of increased accumulation. Folds in the firn stra-tigraphy are difficult to image clearly with conventional GPR pro-cessing methods. Using seismic interpretation methods, wefacilitated structure-oriented filtering by utilizing the firn agemodel to determine the firn structure. In doing so, we furtheredthe application of the IRH theory, which is integral in SMB ana-lyses conducted with radar imagery. This innovation enabled ourinterpretation of deeper (from 16.60 ± 0.04 to 20.15 ± 0.04 m atGTC15) and older (from 1991 ± 31 to 1984 ± 31 days) layersand permitted tuning the age model to a degree of accuracywhich allowed us to derive instantaneous estimates of SMBwhich we averaged annually and multidecadally. Future workwill include application of this methodology to the entire4000 + km GreenTrACS traverse, with independent evaluation atthe 16 core sites. To reduce the labor in interpreting the radarimagery of future work, it would be advantageous to model thefirn age-structure using the kinematic wave equation (Ng andKing, 2011) to capture the advection process imprinted on theradiostratigraphy without having to interpret the Wheeler domainradargram. We picked horizons in the Wheeler domain as anecessary step in applying the structure-oriented filter to theGTC15 Spur West radargram. This interpretive process couldbe avoided by generating the relative age using the kinematicwave equation. Yet, this model requires an independent estimateof firn density and accumulation to satisfy the initial and bound-ary conditions. Deep learning techniques have been recentlyapplied to seismic imaging that automate structure-oriented filter-ing and horizon interpretation problems. By generating syntheticseismograms from numerical structural models as training data(Wu and others, 2020), relative stratigraphic age models havebeen recovered from real seismic data and used for automated iso-chrone horizon interpretation (Geng and others, 2020). The kine-matic wave firn model could serve as a basis for generatingsynthetic radargrams to be used in a deep learning applicationfor historical SMB reconstruction.

Supplementary material. The supplemental material for this article can befound at https://doi.org/10.1017/jog.2020.91

Fig. 9. The GTC15 and MxHL historical SMB for January 1984–January 2017.Uncertainty in GTC15 SMB (± σ) was estimated following Graeter and others (2018).Uncertainties in the MxHL 1984–2017 SMB (± σ) were propagated by Monte Carlosimulations of firn models generated from the parameter distributions of snow dens-ity, 2015–2017 SMB, and MERRA temperature. We applied ± 31 days uncertainty to themeasured ages of isochrones within the simulations.

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Data availability. The 2017 GreenTrACS multi-channel GPR data can befound at https://doi.org/10.18739/A21G0HT84. The MATLAB scripts usedin this analysis are continually under development and can be forked fromthe github repository https://github.com/tatemeehan/GreenTrACS_MxRadar.

Acknowledgments. Greenland Traverse for Accumulation and Climate Studieswas funded by the National Science Foundation Office of Polar Programs: Awards# 1417921 and # 1417678. Additional support of this work was awarded throughthe NASA Idaho Space Grant Consortium Graduate Fellowship and the STEMStudent Employment Program through the U.S. Army Engineer Research andDevelopment Center, Cold Regions Research and Engineering Laboratory. Wethank Steve Arcone and an anonymous reviewer for their recommendationswhich improved the communication of technical aspects of this work andJournal of Glaciology Scientific Editor Shad O’Neel for his additional review ofour manuscript which greatly improved its readability.

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